Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-08T19:27:32.082Z Has data issue: false hasContentIssue false

NWU property of a class of random sums

Published online by Cambridge University Press:  14 July 2016

Jun Cai*
Affiliation:
University of Waterloo
Vladimir Kalashnikov*
Affiliation:
Institute for Information Transmission Problems
*
Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
∗∗Postal address: Institute for Information Transmission Problems, Bol'shoy Karetny 19, 101447 Moscow, Russia

Abstract

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

Type
Short Communications
Copyright
Copyright © 2000 by The Applied Probability Trust 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing: Probability Models. To Begin With, Silver Spring, MD.Google Scholar
Block, H., and Savits, T. (1980). Laplace transforms for classes of life distributions. Ann. Prob. 8, 465474.CrossRefGoogle Scholar
Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 13881402.CrossRefGoogle Scholar
van Harn, K. (1978). Classifying Infinitely Divisible Distributions by Functional Equations (Mathematical Centre Tracts 103). CWI, Amsterdam.Google Scholar
Hansen, B. G., and Frenk, J. B. G. (1991). Some monotonicity properties of the delayed renewal function. J. Appl. Prob. 28, 811821.CrossRefGoogle Scholar
Klefsjö, B. (1982). The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331344.CrossRefGoogle Scholar
Lin, X. (1996). Tail of compound distributions and excess time. J. Appl. Prob. 33, 184195.CrossRefGoogle Scholar
Ross, S. (1983). Stochastic Processes. John Wiley, New York.Google Scholar
Shanthikumar, J. G. (1988). DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397406.CrossRefGoogle Scholar